Littlewood-Paley theorem on an annulus

Question

Suppose a smooth function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ satisfies $$\text{supp}~\hat{f}\subset \{\xi:1<|\xi|<2\}$$ and set functions $f_k$ by saying $$\hat{f_k}:=\hat{f}~\chi_{\{1+2^{-k}<|\xi|<1+2^{-(k-1)}\}}.$$ (You may consider the function $\chi$ as smooth cutoff functions.) Then, is it possible to get the inequality \begin{equation*} ||f||_q\lesssim||(\sum_{k\geq1}|f_k|^2)^{\frac{1}{2}}||_q \end{equation*} for $~q=4$?

Answer 1

Littlewood-Paley theory is not possible on an annulus. If you define a dyadic annulus \begin{equation*} \{ 2^j\leq |\xi|\leq 2^{j+1}\} \end{equation*} rather than dyadic intervals, you run into problems when attempting the $L^p$ extension associated with Littlewood-Paley theory. As you begin the partition of unity and apply Plancherel's theorem to give the $L^2$ isometry between the function $f$ and the Littlewood-Paley operator of $f$, the vector-valued extension/inequality fails because the characteristic function of an $n-$dimensional ball (hence an annulus), $n\geq 2$ is not an $L^p-$multiplier for $p\neq 2$. This result is due to Charles Fefferman.